NP-completeness of combinatorial problems with unary notation for integers

1980 ◽  
Vol 3 (3) ◽  
pp. 397-400
Author(s):  
Martti Penttonen
Keyword(s):  
Polynomial Time ◽  
Knapsack Problem ◽  
Combinatorial Problems ◽  
Np Completeness ◽  
Complete Problems ◽  
Np Complete

Most NP-complete problems remain NP-complete even though the notation for integers is changed to unary. The knapsack problem is an exception, it becomes provably polynomial time recognizable. However, we present a modified knapsack problem that remains NP-complete also in unary notation.

Complexity, computational

2018 ◽  
Author(s):  
Alasdair Urquhart
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Propositional Logic ◽  
Common Type ◽  
Combinatorial Games ◽  
Np Completeness ◽  
Complete Problems ◽  
Np Complete ◽  
Feasible Solutions ◽  
Np Problems

The theory of computational complexity is concerned with estimating the resources a computer needs to solve a given problem. The basic resources are time (number of steps executed) and space (amount of memory used). There are problems in logic, algebra and combinatorial games that are solvable in principle by a computer, but computationally intractable because the resources required by relatively small instances are practically infeasible. The theory of NP-completeness concerns a common type of problem in which a solution is easy to check but may be hard to find. Such problems belong to the class NP; the hardest ones of this type are the NP-complete problems. The problem of determining whether a formula of propositional logic is satisfiable or not is NP-complete. The class of problems with feasible solutions is commonly identified with the class P of problems solvable in polynomial time. Assuming this identification, the conjecture that some NP problems require infeasibly long times for their solution is equivalent to the conjecture that P≠NP. Although the conjecture remains open, it is widely believed that NP-complete problems are computationally intractable.

Looking for Reverse Transformations between NP-Complete Problems

2012 ◽  
pp. 181-206
Author(s):  
Rodolfo A.Pazos R. ◽  
Ernesto Ong C. ◽  
Héctor Fraire H. ◽  
Laura Cruz R. ◽  
José A.Martínez F.
Keyword(s):  
Polynomial Time ◽  
Bin Packing ◽  
Job Scheduling ◽  
Reverse Transformation ◽  
Time Transformation ◽  
Complete Class ◽  
Np Completeness ◽  
Decision Optimization ◽  
Complete Problems ◽  
Np Complete

The theory of NP-completeness provides a method for telling whether a decision/optimization problem is “easy” (i.e., it belongs to the P class) or “difficult” (i.e., it belongs to the NP-complete class). Many problems related to logistics have been proven to belong to the NP-complete class such as Bin Packing, job scheduling, timetabling, etc. The theory predicts that for any pair of NP-complete problems A and B there must exist a polynomial time transformation from A to B and also a reverse transformation (from B to A). However, for many pairs of NP-complete problems no reverse transformation has been reported in the literature; thus the following question arises: do reverse transformations exist for any pair of NP-complete problems? This chapter presents results on an ongoing investigation for clarifying this issue.

scholarly journals Solving the 0/1 Knapsack Problem by a Biomolecular DNA Computer

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Hassan Taghipour ◽  
Mahdi Rezaei ◽  
Heydar Ali Esmaili
Keyword(s):  
Parallel Processing ◽  
Knapsack Problem ◽  
Dna Computing ◽  
Solution Space ◽  
Combinatorial Problems ◽  
Complete Problems ◽  
Short Period ◽  
Np Complete ◽  
Silicon Based ◽  
Dna Computers

Solving some mathematical problems such as NP-complete problems by conventional silicon-based computers is problematic and takes so long time. DNA computing is an alternative method of computing which uses DNA molecules for computing purposes. DNA computers have massive degrees of parallel processing capability. The massive parallel processing characteristic of DNA computers is of particular interest in solving NP-complete and hard combinatorial problems. NP-complete problems such as knapsack problem and other hard combinatorial problems can be easily solved by DNA computers in a very short period of time comparing to conventional silicon-based computers. Sticker-based DNA computing is one of the methods of DNA computing. In this paper, the sticker based DNA computing was used for solving the 0/1 knapsack problem. At first, a biomolecular solution space was constructed by using appropriate DNA memory complexes. Then, by the application of a sticker-based parallel algorithm using biological operations, knapsack problem was resolved in polynomial time.

NONLINEAR COMPUTABILITY BASED ON CHAOS

2000 ◽  
Vol 10 (02) ◽  
pp. 415-429 ◽  
Author(s):  
GABRIELE MANGANARO ◽  
JOSE PINEDA DE GYVEZ
Keyword(s):  
Hamiltonian Path ◽  
Combinatorial Problem ◽  
Combinatorial Problems ◽  
Information Coding ◽  
Computing Power ◽  
Complete Problems ◽  
Np Complete ◽  
Systematic Solution ◽  
Chaotic Simulated Annealing ◽  
Computing Models

Two new computing models based on information coding and chaotic dynamical systems are presented. The novelty of these models lies on the blending of chaos theory and information coding to solve complex combinatorial problems. A unique feature of our computing models is that despite the nonpredictability property of chaos, it is possible to solve any combinatorial problem in a systematic way, and with only one dynamical system. This is in sharp contrast to methods based on heuristics employing an array of chaotic cells. To prove the computing power and versatility of our models, we address the systematic solution of classical NP-complete problems such as the three colorability and the directed Hamiltonian path in addition to a new chaotic simulated annealing scheme.

Method for Organizing Grover's Quantum Oracle

2014 ◽  
Vol 21 (04) ◽  
pp. 1450011
Author(s):  
Hideaki Ito ◽  
Saburou Iida
Keyword(s):  
Quantum Computation ◽  
Knapsack Problem ◽  
Time Complexity ◽  
Quantum Circuit ◽  
Space Complexity ◽  
Complete Problems ◽  
Black Boxes ◽  
Np Complete ◽  
Quantum Oracle ◽  
Toffoli Gates

In a quantum computation, some algorithms use oracles (black boxes) for abstract computational objects. This paper presents an example for organizing Grover's quantum oracle by synthesizing several unitary gates such as CNOT gates, Toffoli gates, and Hadamard gates. As an example, we show a concrete quantum circuit for the knapsack problem, which belongs to the class of NP-complete problems. The time complexity of an oracle for the knapsack problem is estimated to be O(n2), where n is the number of variables. And the same order is obtained for space complexity.

scholarly journals Tuning Frontiers of Efficiency in Tissue P Systems with Evolutional Communication Rules

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
David Orellana-Martín ◽  
Luis Valencia-Cabrera ◽  
Bosheng Song ◽  
Linqiang Pan ◽  
Mario J. Pérez-Jiménez
Keyword(s):  
Polynomial Time ◽  
Efficient Solution ◽  
Efficient Solutions ◽  
Affirmative Answer ◽  
P Systems ◽  
Membrane Systems ◽  
Complete Problems ◽  
Np Complete ◽  
Np Problem ◽  
Computing Models

Over the last few years, a new methodology to address the P versus NP problem has been developed, based on searching for borderlines between the nonefficiency of computing models (only problems in class P can be solved in polynomial time) and the presumed efficiency (ability to solve NP-complete problems in polynomial time). These borderlines can be seen as frontiers of efficiency, which are crucial in this methodology. “Translating,” in some sense, an efficient solution in a presumably efficient model to an efficient solution in a nonefficient model would give an affirmative answer to problem P versus NP. In the framework of Membrane Computing, the key of this approach is to detect the syntactic or semantic ingredients that are needed to pass from a nonefficient class of membrane systems to a presumably efficient one. This paper deals with tissue P systems with communication rules of type symport/antiport allowing the evolution of the objects triggering the rules. In previous works, frontiers of efficiency were found in these kinds of membrane systems both with division rules and with separation rules. However, since they were not optimal, it is interesting to refine these frontiers. In this work, optimal frontiers of the efficiency are obtained in terms of the total number of objects involved in the communication rules used for that kind of membrane systems. These optimizations could be easier to translate, if possible, to efficient solutions in a nonefficient model.

A New Algorithm to Solve the Maximal Connected Subgraph Problem Based on Parallel Molecular Computing

2016 ◽  
Vol 13 (10) ◽  
pp. 7692-7695
Author(s):  
Nan Guo ◽  
Jun Pu ◽  
Zhaocai Wang ◽  
Dongmei Huang ◽  
Lei Li ◽  
...  
Keyword(s):  
Dna Computing ◽  
Combinatorial Problems ◽  
Search Problem ◽  
Connected Subgraph ◽  
Complete Problems ◽  
Portfolio Problem ◽  
Path Search ◽  
Np Complete ◽  
Connected Subgraphs ◽  
Connected Vertex

DNA computing is widely used in complex NP-complete problems, such as the optimal portfolio problem, the optimum path search problem. DNA computing, having the characteristics of high parallelism, huge storage capacity and low energy loss, is very suitable for solving complex combinatorial problems. The maximal connected subgraph problem aims to find a connected vertex subset with maximal number of vertices in a simple undirected graph. Using biological computing technology, we give a new DNA algorithm to solve the maximal connected subgraphs problem with O(n) time complexity, which can significantly reduce the complexity of computing compared with the previous algorithms.

scholarly journals 3-Colorability of Pseudo-Triangulations

2015 ◽  
Vol 25 (04) ◽  
pp. 283-298
Author(s):  
Oswin Aichholzer ◽  
Franz Aurenhammer ◽  
Thomas Hackl ◽  
Clemens Huemer ◽  
Alexander Pilz ◽  
...  
Keyword(s):  
Polynomial Time ◽  
Linear Time ◽  
Plane Graphs ◽  
Np Completeness ◽  
Complete Problem ◽  
Polynomial Time Algorithms ◽  
Complexity Results ◽  
The Family ◽  
General Plane ◽  
Np Complete

Deciding 3-colorability for general plane graphs is known to be an NP-complete problem. However, for certain families of graphs, like triangulations, polynomial time algorithms exist. We consider the family of pseudo-triangulations, which are a generalization of triangulations, and prove NP-completeness for this class. This result also holds if we bound their face degree to four, or exclusively consider pointed pseudo-triangulations with maximum face degree five. In contrast to these completeness results, we show that pointed pseudo-triangulations with maximum face degree four are always 3-colorable. An according 3-coloring can be found in linear time. Some complexity results relating to the rank of pseudo-triangulations are also given.

Sign in / Sign up

Export Citation Format

Share Document